Observations of filaments and filament channels on the Sun indicate that the magnetic fields in these structures exhibit a large-scale organization: filament channels in the northern hemisphere predominantly have axial fields directed to the right when viewed from the positive polarity side of the channel (dextral orientation), while those in the south have axial fields directed to the left (sinistral orientation). In this paper we attempt to explain this pattern in terms of the most natural mechanism, namely, solar differential rotation acting on already emerged magnetic fields. We develop a model of global magnetic flux transport that includes the effects of differential rotation, meridional flow, and magnetic diffusion on photospheric and coronal fields. The model is applied to National Solar Observatory/Kitt Peak data on the photospheric magnetic flux distribution. We also present results from a simulation of solar activity over a period of two solar cycles, which gives a buildup of flux at the poles of a magnitude, in agreement with observations. We find that differential rotation acting on initially north-south oriented polarity inversion lines (PILs) does produce axial fields consistent with the observed hemispheric pattern. The fields associated with switchbacks in the PILs are predicted to have a definite orientation: the high-latitude "lead" arms of the switchbacks are preferentially sinistral (dextral) in the north (south), while the lower latitude "return" arms are, in agreement with observations, preferentially dextral (sinistral). The predicted orientation of fields at the polar crown, however, appear to be in conflict with observations. Further observational studies are needed to determine whether the model can explain the observed hemispheric pattern.